Subject Knowledge and Pedagogical Knowledge

Where will the next generation of UK mathematicians come from? 18 19 March 2005, Manchester www.ma.umist.ac.uk/avb/wherefrom.html The meeting is sponsored by Manchester Institute for Mathematical Sciences and London Mathematical Society Subject Knowledge and Pedagogical Knowledge Doug French, Centre for Educational Studies, University of Hull This is an updated version of a paper that was originally published in Mathematics Education Review in April 2003. It discusses the place of subject knowledge, linked to pedagogical knowledge, in the context of the initial training of secondary school mathematics teachers on PGCE courses. It draws attention to the mathematical weaknesses that students on such courses have and it indicates one way in which the author has tried to grapple with the problem of improving their knowledge of understanding of the mathematics that they are being trained to teach. Introduction At a simple level it would seem to be self-evident that any teacher of mathematics should in some sense 'know' the material that they are being expected to teach. For instance, it would not seem unreasonable that a teacher working with pupils on a GCSE course should themselves be able to obtain near enough full marks on a typical examination paper. It is often naively assumed that somebody who has studied at least some mathematics at degree level would be able to achieve that with little difficulty, although there may be the occasional topic that is unfamiliar. Again it might be assumed that somebody used to working at a higher level would be able to acquire the necessary knowledge readily. It is very clear to those who work with students aspiring to be secondary teachers that many certainly do not have this level of knowledge in relation to A level mathematics, but there are often very significant gaps and misunderstandings with much more elementary aspects of the subject. This is even more evident amongst students training to be primary teachers. Formal levels of subject qualification are not sufficient to ensure that a potential teacher has an appropriate level of subject knowledge, even when it is viewed in this restricted way. However, the ability to obtain a mark of 100% on an examination paper, whilst necessary, is far from sufficient for effective teaching. A teacher's ability to prepare effective lessons and to respond perceptively and flexibly to the multitude of difficulties that pupils encounter with mathematics is dependent on their own depth of understanding of the topics involved and their own powers of mathematical thinking, as well as their more general pedagogical skills and understandings. Even in the context of a narrow examination defined curriculum, there is always the need to prepare students for longer term examination goals as well as the immediate goal. However, the aims of a mathematical education are much wider than helping pupils to pass examinations. Subject knowledge which embraces depth of understanding, an ability to think mathematically and subject related pedagogical knowledge, as well as content knowledge at an appropriate level, is vitally important to all who teach mathematics. What is subject knowledge and why is it important? The phrase 'subject knowledge' is used in the recently revised Standards for Initial Teacher Training, DfES (2002), which sets out what is expected of students in order to achieve

Qualified Teacher Status. An appendix setting out the requirements for mathematics in some detail, that had appeared in an earlier version of the Standards, DFEE (1998), was omitted in the 2002 version. One of the most telling subject knowledge requirements in the 1998 version of the Standards stated that those to be awarded Qualified Teacher Status must be able to 'cope securely with subject-related questions which pupils raise'. This very demanding, but very appropriate, requirement was omitted in the 2002 revision of the Standards. Of course, it was not at all easy to assess in any direct and meaningful way, but it certainly suggested a deeper and wider knowledge than is required to get full marks on an examination paper. Given the opportunity some pupils ask very demanding questions about ideas that are well within the school curriculum and all pupils should certainly be encouraged to be curious about mathematics and to probe deeply. Teachers need wide and deep knowledge if they are to respond well, even though the response may often be to ask further questions and to point to ways of finding out or exploring further. Moreover, teachers need subject knowledge that is linked closely to pedagogical knowledge. For example, an awareness of common misconceptions and ways of looking at them, the importance of forging links and connections between different mathematical ideas and the flexibility that comes from seeing alternative ways of looking at the same idea or problem are all essential for effective teaching. There is a very wide gulf between a desirable level of subject knowledge and the level of knowledge that most student teachers display either at the start or in many cases at the end of their course. This gulf is brought out very dramatically in Ma Liping's comparative study of the subject knowledge of elementary school teachers in China and the United States of America, Ma (1999). She posed four simple arithmetical problems to a sample of teachers in each country and looked at their responses in terms of how they themselves would solve the problem and how they would approach it with pupils in the classroom. For example, one of the problems was 1 3 4 1 2. Only 9 out of 21 American teachers could answer the question correctly, whereas all of a sample of 72 Chinese teachers were successful. Moreover the successful American teachers were much less successful than their Chinese counterparts in explaining why the process worked or in finding examples to exemplify the calculation. Ma notes also that the American teachers had spent a longer period in higher education before qualifying compared to those from China, including study of mathematics at a higher level. One cannot help but reflect sadly that teachers in the UK are more likely to reflect the weaknesses of the USA than the strengths of China and, of course, that the performance of Chinese pupils is better than those in the UK and the USA. Another study which relates to this issue is Effective Teachers of Numeracy, Askew et al (1997), which investigated the teaching styles of primary school teachers and identified the characteristics of those who were most effective as measured by improvements in pupil performance. The teachers fell into three broad orientations, referred to as 'transmission', 'discovery' and 'connectionist'. Most teachers displayed facets of more than one of these orientations, but usually one tended to be dominant. The 'transmission' orientation is characterised by a traditional 'explain and practise' style and 'discovery' emphasises setting tasks through which pupils discover ideas for themselves. The 'connectionists' were teachers who put a lot of emphasis on drawing out the connections between mathematical ideas and developing understanding through discussion and it was teachers with this as their dominant orientation who were found to be most effective. There appeared to be no connection between the level of the teachers' formal mathematical qualifications and their effectiveness. Both these studies seem to suggest that high level mathematical qualifications are much less important than the depth of teachers' understanding and their ability to make connections within

the mathematics of the school curriculum. Study of mathematics beyond school level might be expected to reinforce understanding of elementary mathematics, but anecdotal evidence suggests that this does not necessarily happen. There is little or no formal research evidence relating to the subject knowledge of secondary school teachers, but it would not be surprising if replications of the two types of study produced similar results. If the quality of mathematical education is to be improved, the emphasis in the Standards on subject knowledge linked closely to pedagogical knowledge is entirely appropriate for our student teachers. The key issue for all involved in teacher education is how to extend the subject knowledge of our students, who often have a very narrow topic and technique oriented view of the subject with limited understanding of where ideas come from, how they are linked and how they can be applied to solve unstructured problems or to generate proofs. They are largely products of a system which has encouraged a narrow view of the subject and they will themselves go on to reinforce that view in their own teaching if their vision is not extended and their understanding and knowledge enhanced. How can the problem be tackled? Formal subject qualifications and narrow testing or auditing procedures are insufficient to determine whether student teachers have acquired this more broadly defined subject knowledge. Assessment needs to be placed in the context of tasks designed to enhance subject knowledge, where advice can be offered to help students extend and deepen their knowledge. Formative assessment is much more productive than summative assessment with student teachers just as it is with pupils. The excellent sequence of three short articles by Wiliam (1999/2000), draws attention, in the context of school pupils' learning, to the idea of 'rich questions' as a way of revealing misconceptions and to the virtues of offering comments without marks or grades. These ideas seem to me to apply to the way in which subject knowledge should be approached with those aspiring to Qualified Teacher Status. One of my approaches to improving my secondary PGCE students' subject knowledge has been to develop a Subject Knowledge Workbook. This consists of a variety of tasks designed to enhance both mathematical and pedagogical knowledge. I describe below some of these tasks, their rationale and the sort of responses that students make, but it is important first to describe how the Workbook is used and how this is linked to the ideas of formative assessment espoused by Wiliam. At intervals during the one year course I ask students to hand in their Workbook completed up to a certain page. The books are returned to them with comments indicating areas where further work is needed, together with occasional hints and suggestions, and praise for some good comments or ideas. I do not give them a mark, although I do indicate by means of a red tick at the bottom of each page when I think that sufficient work has been done on that page's topic. As well as the written comments students are given oral feedback, both individually and as a group, and they can always ask for additional advice from myself, their mentors or anybody else. The Workbooks are a part of the formal course assessment in that they have to be handed in complete with their final assignment and have to be such that each page is worthy of a red tick! They do not in any way contribute to the grade that is awarded for method work. Whilst the Workbook seems to me only to be scratching at the surface of a very big problem, it is regarded by the students as one of the most valuable things that they do during the university based part of the course. They commonly say that they find it both enjoyable and challenging: 'it makes you think' is a common reaction. Our last Ofsted inspector was very suspicious at first, not least because I argued that the Workbook took the place of a subject audit and was much

more valuable because it grasped the problem of doing something about the deficiencies that an audit might reveal, besides attempting to do much more. The final report on Hull's mathematics course, Ofsted (2000) said: 'The university has recently developed a very useful 'Subject Knowledge Workbook', which is designed to enhance trainees' subject knowledge and highlight common errors and misconceptions. Trainees make good use of this workbook and their progress is informally reviewed and well supported by the method tutor who ensures that training is differentiated to build on previous academic experience.' So, there is life without a more conventional audit! What aspects of mathematics and mathematics learning are particularly significant? There is a lot that I could say about the difficulties that my students encounter in working on their Subject Knowledge Workbooks. I will highlight three areas by commenting on questions taken from the Workbook: Explaining standard procedures and finding alternatives; Discussing conceptual difficulties; Solving problems and creating proofs in geometry. 1. Suggest various ways of helping pupils understand that: 2 3 = 8. 3 4 9 It is rare to find students who can readily explain where the 'turn it upside down and multiply' procedure for division of fractions comes from or who can offer any alternative procedures. Their first reaction is often 'I have never been told why it works' or 'I have never been taught any other way of doing it'. As they become attuned to expectations they begin, often painfully, to think about mathematical ideas from first principles by trying to put themselves in the place of a pupil who is challenged to find a way to divide one fraction by another when they have not previously encountered a formal procedure. Seeing that it means 'How many times does 3 4 go into 2 3?' is often a valuable first step and thinking about an estimate for the answer is often useful. Interesting alternatives often emerge besides the more conventional explanations. I give two here: Multiply both fractions by 12: 2 3 3 4 = 8 9 = 8 9. Solve 3 4 x = 2 3 : 4 3 3 4 x = 2 3 4 3 x = 8 9. 2. Give several alternative methods to show that: 2 < 3 5 7 This example has often resulted in incorrect reasoning when students have 'cross multiplied' and obtained 14 < 15. They then argue that because that is true the original result must hold. This is, of course, false logic, because they have started by assuming that what they are trying to prove is true. Using the same logic I can prove that 1 = 2, by multiplying both sides by zero and

noting that 0 = 0, which is certainly true. The reasoning is clearly incorrect. The question could be rephrased to ask which fraction is larger, but teachers should be able to reason correctly, so I shall not change it! 3. Criticise this statement: a week is 7 days, so w= 7d, where w stands for week and d stands for day. 4. Comment on the misconception revealed by a pupil whose response to 'simplify k+ 2p+ 4 is 3k+ 4= 7k. Both these examples illustrate the erroneous use of 'letters as objects', a misconception which I always discuss with students at length when we consider the problems of learning algebra. In both cases, substituting numbers will show that the statements are incorrect, but that is not usually students' immediate response. Indeed, in the first case, they do not always realise that there is anything wrong. Misconceptions of this kind are commonly reinforced by the approaches to algebra in many current school textbooks, a problem that I discuss at length in the second chapter of my book Learning and Teaching Algebra, French (2002). The current version of the National Curriculum, DfEE/QCA (1999), has accorded greater importance to aspects of geometrical reasoning and proof, something which many students acknowledge did not feature strongly in their own school experience. They are aware of key theorems, but they often find it very difficult to apply them to problems or use them in proofs. The difficulty with many geometrical problems is to see a key feature of the diagram or to add a line or two which enables a key result to be applied. The two examples from the Workbook which follow illustrate the importance of seeing the right step, which raises the fascinating issue of how appropriate strategies might be acquired. 5. Two circles intersect at P and Q. PA and PB are diameters. Prove that the points A, Q and B lie in a straight line. P B A Q 6. Give a proof (not using vectors) that the medians of a triangle meet in a point which divides each median in the ratio 2 to 1.

C Q P A R B In example 5, students readily draw in the line segments AQ and BQ, but do not always then see that including PQ and applying the angle in a semi-circle theorem gives an easy solution. In example 6, I have recently inserted not using vectors, because a vector solution used to be the favourite method. Whilst a vector approach is of interest, it really is 'using a sledgehammer to crack a nut', because there are shorter, simpler methods which are accessible to younger pupils, and simplicity is something to encourage. In spite of dropping large hints by looking at appropriate strategies when discussing other problems in the weeks before they encounter this problem in the Workbook many students find it extraordinarily difficult to find a simple proof by, for instance, ignoring (or deleting) the line CR, drawing in the line PQ and looking at similar triangles. What are the ways forward in a wider context? As I have suggested earlier, the subject knowledge requirements of the Standards could be seen to be very demanding, but there is a very great latitude in the way that they are being interpreted, particularly in relation to the evidence higher education institutions (and the providers of school based routes to Qualified Teacher Status) expect tutors to produce for their own internal purposes or for Ofsted inspectors, who sometimes have a rather narrow content based focus. In both cases we are up against the obsession that the system has at all levels with summative assessment, which runs counter to all the evidence that formative assessment (see Wiliam (1999/2000) is much more effective in raising standards. We must constantly counter the arguments for more testing by providing evidence that there are other much more effective ways of raising standards, both in teacher education and in school mathematics. A major difficulty that teacher training providers face is the problem of time. Students are in school for two thirds of a secondary PGCE course and for a half of a primary PGCE course, where the subject knowledge demands are spread across a whole range of subjects. With school based training routes, students spend almost all their time in schools. It is very difficult for school based time to be used to enhance subject knowledge except in an incidental way, because the daily demands of coping with lesson planning, classroom management and pupil behaviour are inevitably going to be dominant. Self study has an important part to play, but there is also a vital need for discussion and reflection, both at an individual level and within wider groups, and for access to the expert knowledge which PGCE subject tutors should surely have. There are many legitimate competing demands for time within a PGCE course. I would suggest that the balance between time spent in school and university is skewed too much towards school at present, but that is a subject for another paper! Clearly the problem of broadly defined subject knowledge is wider than the needs of those aspiring to Qualified Teacher Status. We know only too well that there is a dire shortage of well qualified teachers and that there are many who are teaching mathematics, whether nominally qualified to do so or not, who would readily admit to having considerable deficiencies in their subject knowledge, as well as many others who have deficiencies, often at a very elementary level, of which they are sadly not aware. There is nothing new about calls for more in-service

training for mathematics teachers, but the need for more will continue to be urgent for the foreseeable future. The National Strategies have resulted in a substantial increase in training for teachers, but the emphasis has not primarily been on subject knowledge, although the focus on pedagogy inevitably impinges on the issues discussed in this paper. However, we have little evidence about the effectiveness of different forms of in-service training. It may, for instance, be much more effective for a few teachers to do longer intensive courses, like the 20 day courses for primary teachers a few years ago, than large numbers of teachers doing very short courses. We can always dream about the golden age that may arrive at some point in the future. We must constantly strive towards that and not lose a vision of how things could be, but we do have to identify our priorities for tomorrow and the next day, by finding ways - often small ways - to improve things whilst working with what we have. I am constantly amazed at the goodwill and enthusiasm of so many student teachers, who often survive in very trying circumstances and still manage to smile. In spite of all the difficulties we can continue to make a difference. Bibliography Askew, M., Brown, M., Rhodes, V., Johnson, D., Wiliam, D. (1997) Effective Teachers of Numeracy King's College, London DfEE (1998) Standards for the Award of Qualified Teacher Status (Annex A of DfEE Circular 4/98) London: DfEE DfEE/QCA (1999) The National Curriculum for England: Mathematics London: DfEE/QCA DfES (2002) Qualifying to Teach: Professional Standards for Qualified Teacher Status and Requirements for Initial Teacher Training London: DfES French, Doug (2002) Learning and Teaching Algebra London: Continuum Ma, Liping (1999) Knowing and Teaching Elementary Mathematics Mahwah, NJ: Lawrence Erlbaum Ofsted (2000) Secondary Subject Inspection 1999/2000: University of Hull (Mathematics) London: Ofsted Wiliam, Dylan (1999/2000) 'Formative Assessment in Mathematics' Equals, Vol. 5, no. 2 and no.3 and Vol.6, no. 1 A Free Offer I have a limited stock of old copies of the Subject Knowledge Workbook which I will willingly send to anybody on request. I am very happy for anybody to make use of any of the material in the Workbook with their students in whatever form seems appropriate. Centre for Educational Studies, University of Hull, Hull, HU6 7RX d.w.french@hull.ac.uk